cotangent bundle


Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




  • (shape modality \dashv flat modality \dashv sharp modality)

    (ʃ)(ʃ \dashv \flat \dashv \sharp )

  • dR-shape modality\dashv dR-flat modality

    ʃ dR dRʃ_{dR} \dashv \flat_{dR}

  • tangent cohesion

    • differential cohomology diagram
    • differential cohesion

      • (reduction modality \dashv infinitesimal shape modality \dashv infinitesimal flat modality)

        (&)(\Re \dashv \Im \dashv \&)

      • graded differential cohesion

        • fermionic modality\dashv bosonic modality \dashv rheonomy modality

          (Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

        • id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous *

          \array{ && id &\dashv& id \ && \vee && \vee \ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \ && \bot && \bot \ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \ && \vee && \vee \ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \ && \bot && \bot \ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \ && \vee && \vee \ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \ && \bot && \bot \ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \ && \vee && \vee \ && \emptyset &\dashv& \ast }



          Lie theory, ∞-Lie theory

          differential equations, variational calculus

          Chern-Weil theory, ∞-Chern-Weil theory

          Cartan geometry (super, higher)



          Given a differentiable manifold XX, the cotangent bundle T *(X)T^*(X) of XX is the dual vector bundle over XX dual to the tangent bundle TxT x of XX.

          A cotangent vector or covector on XX is an element of T *(X)T^*(X). The cotangent space of XX at a point aa is the fiber T a *(X)T^*_a(X) of T *(X)T^*(X) over aa; it is a vector space. A covector field on XX is a section of T *(X)T^*(X). (More generally, a differential form on XX is a section of the exterior algebra of T *(X)T^*(X); a covector field is a differential 1-form.)

          Given a covector ω\omega at aa and a tangent vector vv at aa, the pairing ω,v\langle{\omega,v}\rangle is a scalar (a real number, usually). This (with some details about linearity and universality) is basically what it means for T *(X)T^*(X) to be the dual vector bundle to T *(X)T_*(X). More globally, given a covector field ω\omega and a tangent vector field vv, the paring ω,v\langle{\omega,v}\rangle is a scalar function on XX.

          Given a point aa in XX and a differentiable (real-valued) partial function ff defined near aa, the differential d af\mathrm{d}_a f of ff at aa is a covector on XX at aa; given a tangent vector vv at aa, the pairing is given by

          d af,v=v[f], \langle{\mathrm{d}_a f, v}\rangle = v[f] ,

          thinking of vv as a derivation on differentiable functions defined near aa. (It is really the germ at aa of ff that matters here.) More globally, given a differentiable function ff, the de Rham differential df\mathrm{d}f of ff is a covector field on XX; given a vector field vv, the pairing is given by

          df,v=v[f], \langle{\mathrm{d}f, v}\rangle = v[f] ,

          thinking of vv as a derivation on differentiable functions.

          One can also define covectors at aa to be germs of differentiable functions at aa, modulo the equivalence relation that d af=d ag\mathrm{d}_a f = \mathrm{d}_a g if fgf - g is constant on some neighbourhood of aa. In general, a covector field won't be of the form df\mathrm{d}f, but it will be a sum of terms of the form hdfh \mathrm{d}f. More specifically, a covector field ω\omega on a coordinate patch can be written

          ω= iω idx i \omega = \sum_i \omega_i\, \mathrm{d}x^i

          in local coordinates (x 1,,x n)(x^1,\ldots,x^n). This fact can also be used as the basis of a definition of the cotangent bundle.


          Symplectic structure

          Every cotangent bundle T *XT^\ast X carries itself a canonical differential 1-form

          θΩ 1(T *X) \theta \in \Omega^1(T^* X)

          with the property that under the isomorphism

          j:Γ(T *X)Ω 1(X) j \;\colon\; \Gamma(T^* X) \stackrel{\simeq}{\to} \Omega^1(X)

          between differential 1-forms and smooth sections of the cotangent bundle we have for every smooth section σΓ(T *X)\sigma \in \Gamma(T^* X) the identification

          σ *θ=j(σ) \sigma^* \theta = j(\sigma)

          between the pullback of θ\theta along σ\sigma and the 1-form corresponding to σ\sigma under jj.

          This unique differential 1-form θΩ 1(T *X)\theta \in \Omega^1(T^* X) is called the Liouville-Poincaré 1-form or canonical form or tautological form on the cotangent bundle.

          The de Rham differential ωdθ\omega \coloneqq d \theta is a symplectic form. Hence every cotangent bundle is canonically a symplectic manifold.

          On a coordinate chart n\mathbb{R}^n of XX with canonical coordinate functions denoted (x i)(x^i), the cotangent bundle over the chart is T * n n× nT^\ast \mathbb{R}^n \simeq \mathbb{R}^n \times \mathbb{R}^n with canonical coordinates ((x i),(p j))((x^i), (p_j)). In these coordinates the canonical 1-form is (using Einstein summation convention)

          θ=p idx i \theta = p_i d x^i

          and hence the symplectic form is

          ω=dp idq i. \omega = d p_i \wedge d q^i \,.

          Last revised on November 23, 2017 at 07:48:04. See the history of this page for a list of all contributions to it.